A Collection of Mathematical Models for Dispersion in Surface Water and Groundwater

Manuscript Completed: August 1981 Date Published: June 1982

R. B. Codell, U.S. Nuclear R_egulatory Commission K. T. Key, Pacific Northwest Laboratory G. Whelan, Pacific Northwest Laboratory

Division of Engineering Office of Nuclear Reactor Regulation U.S. Nuclear Regulatory Commission Washington, D.C. 20555

NUREG-0868

• !••

• J,,'."

ABSTRACT

This report represents a collection of some of the manual procedures and simple computer programs used by the Hydro.logic Engineering Section for com-puting the fate of routinely or accidentally released radionuclides in surface water and groundwater. All models are straightforward simulations of disper-sion with constant coefficients in simple geometries •

•

i i i

•

CONTENTS

ABSTRACT •

1.0 INTRODUCTION

2.0 SURFACE WATER MODELS

2.1 PROGRAM STTUBE - DILUTION AND TRAVEL TIMES IN A STEADY-STATE RIVER •

2.1.1 Introduction •

2.1.2 Theory •

2.1.3 Model Applications

2.1.4 Program STTUBE - User Documentation

2.1.5 Sample Problem

2.1.6 Model Confirmation.

..

2 .2 PROGRAM TUBE - ESTIMATION OF THE STREAM-TUBE LOCATIONS AND THE DISPERSION FACTORS

2.2.1 Introduction •

iii

1.1

2.1

2.1

2.1

2.1

2.5

2.7

2 .11

2.13

2.24

2.24

2.2.2 Theory • 2.24

2.2.3 Computation of Cumulative Flow and Dispersion Factor • 2.33

2.2.4 Program TUBE - User Documentation 2.37

2.2.5 Sample Problems 2.39

2.3 PROGRAM RIVLAK - CONCENTRATION IN A RIVER OR NEAR-SHORE REGION OF A LARGE LAKE FROM A NONSTEADY SOURCE 2.40

2.3.1 Introduction •

2.3.2 Theory •

2.3.3 Model Application

l.s .4 Program RI\ll-AK- ~--User Doc.ume_ntation 2.3.5 Sample Problem

v

2.40

2.43

2.47

2.52

2.56

2.3.6 Model Confirmation 2.57

REFERENCES 2~63

3.0 GROUNDWATER MODELS . 3.1 3.1 GENERAL DEVELOPMENT • 3.1

3 .1.1 Introduction • 3.1

3 .1.2 Theoretical Development. 3.2

3 .1.3 Point Concentration Model 3.5

·.3 .1.4 Flux Model 3.9

3 .1.5 Generalization of Instantaneous Models 3.11

3 .1.6 Estimation of Coefficients and Parameters . 3 .11 '

3.1.7 Radiological Decay 3.13

3.1.8 Evaluation of Daughter Product Movement 3.15

3.1.9 Screening 3.18

3.2 PROGRAM GROUND - THREE-DIMENSIONAL GROUNDWATER CONCENTRATION AND FLUX TO A RIVER FROM A NONSTEADY POINT OR LINE SOURCE. 3.19

3.2.1 Introduction • 3.19

3.2.2 Point Concentration Model 3.19

3.2.3 Flux Model 3.21

3.2.4 Model Applications 3.22

3.2.5 Program GROUND - User Documentation 3.24

3.2.6 Sample Problems 3.29

3.2.7 Model Confirmation 3.30

3.3 PROGRAM GRDFLX GROUNDWATER CONCENTRATION AND FLUX TO POINTS DOWNGRADIENT FROM AN AREA SOURCE 3.39

3.3.1 Introduction • 3.39

3.3.2 GRND Model 3.40

vi

3.3.3 FLUX Model

3.3.4 Model Applications

3.3.5 Program GRDFLX - User Documentation I

3 .3.6 Sample Problem

3.3.7 Model Confirmations

REFERENCES

4.0 SIMPLE MANUAL CALCULATIONS

4.1 INTRODUCTION

4.2 SURFACE-WATER METHODS

4.2.1 Well-Mixed Region or Averaged Concentration Across River from a Continuous Source •

4.2.2 Sectionally Averaged Peak Concentration from an Instantaneous Discharge •

4.2.3 Concentration in a River or Lake Close to a Continuous Source •

4.2.4 On a River or Lake Close to Source for Instantaneous Point Source •

4.2.5 On a River in Intermediate Region - Onshore Release and Intake.

4.3 GROUNDWATER-SURFACE-WATER INTERFACE -INSTANTANEOUS SOURCE.

4.4 DILUTION AND TRAVEL TIME AT DOWNGRADIENT WELLS FOR AN INSTANTANEOUS SOURCE •

4.4.1 Vertically Mixed Region.

4.4.2 Unmixed Region

4.4.3 Intermediate Region

4.4.4 Travel Time •

APPENDIX A - PROGRAM STTUBE

APPENDIX B - PROGRAM TU-BE

vii

3.41

3.42

3.45

3.59

3.60

3.73

4.1

4.1

4.1

4.2

4.2

4.3

4.3

4.4

4.9

4.11

4.12

4.12

4.13

4.13

A.1

B.1

APPENDIX C - PROGRAM RIVLAK

APPENDIX D - PROGRAM GROUND

APPENDIX E - PROGRAM GRDFLX

APPENDIX F - PROGRAM GRDFLX SAMPLE DATA LISTING

APPENDIX G - GRDFLX SAMPLE SIMULATION RESULTS.

APPENDIX H - SOUTH FARMINGDALE CHROMIUM DUMP SITE DATA LISTING

viii

C.l

D.1

E.1

F.1

G.l

H.1

FIGURES

2.1 Model of an Infinitesimal Stream-Tube in a Natural Stream 2.2

2.2 Dimensionless Concentration of Nondecaying Radionuclide Discharge at the Shoreline Versus Dimensionless Downstream Distance. 2.6

2.3 Jet Discharge Approximated by a Line Source • 2.13

2.4 Sample Problem - Program STTUBE 2.14

2.5 Normalized Concentration at 3 Points Across River Example 2.18

2.6a Distribution of Dye Concentration with Respect to Normalized Cumulative Discharge, Transects 3 and 5. - 2.20

2.6b Distribution of Dye Concentration with Respect to Normalized Cumulative Discharge, Transects 7, 9, and 10. 2.21

2.7a Distribution of Dye Concentration with Respect to Normalized Cumulative Discharge, Transects 3 and 5 2.22

2.7b Distribution of Dye Concentr.ation with Respect to Normalized Cumulative Discharge, Transects 7, 9, and 10 2.23

2.8 Correlation of Channel Depth Versus Flow Data 2.27

2.9 Correlation of Factor a • 2.32

2.10 Approximate River Cross Section

2.11 Cumulative Flow Computed by Eq. (2.24) •

2.12 Sample Problem 1 - Program TUBE

2.13 Sample Problem 2 - Program TUBE

2.14 Geometry of Simple 2-D River Model

2.15 Near-Field Model

2.16 Arbitrary Source Release

2.17 Sample Problem - Program RIVLAK (River)

-2 .18 Sample P-F-oblem - P--rogr-am RlVLAK (Lake) •

ix

2.33

2.34

2.41

2.42.

2.44

2.47

2.54

2.58

2.59

2.19a Transverse Distribution of Dye Concentration, Transects 3 and 5

2.19b Transverse Distribution of Dye Concentration, Transects 7, 9, and 10

3.1 Idealized Groundwater System for Point Concentration Model, Point Source.

3.2 Idealized Groundwater System for Point Concentration Model, Horizontal Line Source.

3.3 Vertically Averaged Groundwater Dispersion Model •

3.4 Groundwater-Surface-Water Interface, Flux Model

3.5 Concentration of 230Th as a Function of Distance from Trench

3.6 Arbitrary Source Release.

3.7 Sample Problem - Program GROUND (Point Concentration) •

3.8 Sample Program - Program GROUND (Flux) •

3.9 Tritium Releases to Groundwater from ICPP Deep Injection Wells - curies/year.

3.10 Model - Prototype Comparison for Tritium Concentration in Picocuries/Milliliter - October 1961

3.11 Model - Prototype Comparison for Tritium Concentration in

2.61

2.62

3.6

3.7

3.8

3.9

3.17

3.27

3.31

3.33

3.34

3.37

Picocuries/Milliliter - July 1966 . 3.38

3.12 Illustration of the Leaching Function and Related Components 3.44

3.13 Example Tabular Source Function Distribution. 3.52

3.14 11 Measured 11 Areal Extent of Contaminatfon 3.62

3.15 Source Function for: (a) Cadmium and (b) Chromium 3.64

3.16 Temporal Variation in the Chromium Concentration at Different Locations Downgradient of the Area Source 3.67

3.17 Spatial Variations in the Chromium Concentration for the Years 1949, 1953, 1958, and 1962

3.18 Temporal Variation in the Cadmium Concentration at Different

3.68

Locations Downgradient of the Area Source 3.69

x

3.19 Spatial Variation in the Cadmium Concentration for the Years 1949, 1953, 1958, and 1962

4.1 Mixing Factor •

TABLES

2.1 Computation of Parameters for Stream-Tube Model Example

2.2 Dispersion Factors (x 103), ft5/sec2, by Program TUBE .

2.3 Linearized Source Release

3.1 Characteristic Travel Length for Daughters of 238u

3.2 Concentration of 226Ra Preceding 230Th Peak at t = 710,000 Years

3.3 Linearized Source Release

3.4a Output Description •

3.4b Output Description •

3.4c Output Description .

3.5 Sample Problem Data-Porous Media Characteristics •

3.6 Sample Problem Data-Radionuclide Characteristics .

3.7 Data for the South Farmingdale Chromium-Cadmium Site Modeling •

xi

3 .70

4.6

2.12

2.24

2.55

3.16

3.18

3.27

3.56

3.57

3.58

3.60

3.61

3.66

LIST OF SYMBOLS

(Please refer to user documentation for units used.)

A = River cross-sectional area B = River width b =Width of an area source, or length of a horizontal line source C = Radionuclide concentration

Ci = Concentration at a point due to an instantaneous 1 curie release Ch = Chezy coefficient

D = Dispersion factor D . th d . l . d d f t i = i ra ionuc 1 e ose ac or DL = Dilution Dx = Dispersion factor that varies in the directiun of flow d = Stream depth, also aquifer thickness g = Acceleration of gravity G =Concentration in liquid phase in the nonflowing voids

Ex = Longitudinal turbulent Qispersion coefficient E = Lateral turbulent dispersion coefficient y F = Flux across a plane

F. =Flux due to an instantaneous 1 curie release 1

H = Constant aquifer thickness h = Depth of aquifer or piezometric level i =Hydraulic gradient in aquifer K = Dimensionless constant k =Permeability

Kd = Distribution coefficient kx = x-component hydraulic conductivity Lc = Characteristic distance for a daughter product

i = Length of a horizontal area source Lr = Leach rate M = Total curies under the release function curve

Mmax = Number of truncated terms in an infinite series

xiii

n = Manning coefficient or total porosity in groundwater flow n = Sectional mean Manning coefficient

ne = Effective porosity P = Wetted perimeter q = Cumulative discharge

q* = Concentration associated with the nonflowing portion of the volume ' q = Dimensionless cumulative discharge

Q = Total river flow r = Radial distance

Rh = Hydraulic radius Rc = Radius of curvature Rd = Retardation factor Rs = Real specific gravity S = Energy slope

Ss = Source strength S(t) = Distributed sink

t = Time T = Transmissivity = kH/ne

t 1,t2,t3 =Time range of integration t 112 =Half-life of radionuclide

Ur = Radial velocity u = Stream-ve)ocity or undirectional groundwater velocity u = Sectional mean velocity

uf = Net downstream (freshwater) velocity u* = Shear velocity V = Volume v = Scalar velocity

W(t) = Distributed source X(x,t) =Green's function in the x-direction Y(x,t) =Green's function in the y-dir~ction Z(x,t) =Green's function in the z-direction

x = Distance downstream x = Dimensionless distance downstream

xiv

y = Distance across river or lateral distance in an aquifer z = Vertical distance

zs = Vertical position of a source Ys = y-coordinate of the source ax = x-component of dispersivity ay = y-component of dispersivity az = z-component of dispersivity Pb = Aquifer bulk density a = Empirical coefficient

ax = Empirical coefficient ay = Empirical coefficient

y =Arbitrarily large number, typically 20-200 A = Decay coefficient, = (ln 2}/half life ~ = Factor characterizing degree of sectional mixing x = Nondecaying concentration x = Dimensionless concentration

xv

1.0 INTRODUCTION

This report is a collection of some of the manual procedures and simple BASIC and FORTRAN language computer programs used by the Hydrologic Engineering Section(a) (HES) for computation of ihe fate of radionuclides

which enter surface water or groundwater through routine or accidental releases. Models are presented for rivers, the Great Lakes (near-shore zone), and groundwater. All models are straightforward simulations of dispersion with constant coefficients and, in general, simple geometries.

Chapter 2.0 deals with surface-water models. Programs STTUBE and TUBE are useful for two-dimensional dispersion of a continuous source into a river after steady state has been attained. These programs require actual river cross sections and roughness coefficients. Program RIVLAK is also for disper-sion in a river, but the source may be either steady or unsteady. The river channel must be of constant width and depth. In addition, program RIVLAK can be used for computing the two-dimensional dispersion in the near-shore regime of the Great Lakes or other large lakes.

All surface-water models neglect the effect of uptake of radionuclides on sediments. Neglecting this effect is a conservative assumption in most cases.

Chapter 3.0 describes the groundwater dispersion models. Program GROUND is used for calculating the dispersion in a three-dimensional aquifer and is most useful for determining the concentration at wells downgradient of a source released from a vertical plane. Program GROFLX provides the same function, except it considers a horizontal area source. Option is provided for calculat-ing the eventual release of contaminated groundwater to a downgradient inter-cepting surface-water body.

Chapter 4.0 describes manual methods which may be used to estimate dilu-tions in surface water and groundwater, without the need of computer programs. These methods are all that are required in many cases, or they may be used as initial estimates before the more thorough computer programs are employed.

(a) Hydrologic Engineering Section, Hydrologic and Geotechnical Engineering Branch, Division of Engineering, Office of Nuclear Reactor Regulation, U_.S. Nuclear Regulatory Commission.

1.1

Each chapter and program description is written to stand alone. Revisions and new programs will become available on an irregular basis.

The users of these models must be cautioned that no guarantees of appli-cability or conservatism are expressed or implied. In most cases, personal judgement and experience must be relied upon to determine the applicability of a model in a given situation. It should be emphasized that these models should be used as tools for conta_mination assessment; a variety of scenarios should be addressed. Var~ing parameters to obtain ranges--as opposed to fixed quan-tities--is highly encouraged.

The user should also note that the number of significant digits varies from computer to computer, depending on the word length used. Hence, modeling results should be read with this in mind. Because many of the machine-dependent advanced features of the languages are not employed, the transfer-ability of the programs is ensured.

These models and methods are provided as useful tools for radionuclide migration analyses. Use of these methods does not automatically ensure NRC approval, nor are they required procedures for nuclear facility licensing. Furthermore, by publishing this guidance, NRC does not wish to discourage independent assessments or furtherance of the state of the art.

1.2

2.0 SURFACE WATER MODELS

2 .1 PROGRAM STTUBE - DILUTION AND TRAVEL TIMES TNJX--STEAUY-=STATE ___ RIVER

2.1.1. Introduction

Program STTUBE is useful for dispersion computations in unidirectional rivers with varying cross sections. Computations are performed in 11 stream-tube11 coordinates, in which complex river cross sections are mapped into a new river discharge-based coordinate system so that their mathematical represen-tation can be simplified. The relationship between stream-tube coordinates and geometric coordinates is developed in Section 2.2 and is computed with program TUBE. The method has been used successfully to simulate the disper-sion of both conservative and nonconservative substances in a number of rivers. (l,2)

Application of this model is restricted to those portions of the river removed from influences of the discharge. Initial dilution near the point of discharge is often controlled by turbulent mixing induced by near-field momentum effects such as jet discharges. Procedures for near-field computa-tions are described in Reference 8.

2.1.2. Theory

For nontidal rivers, the flow is assumed to be uniform and approximately steady. Under these conditions, the dispersive transport in the flow direc-tion may be neglected compared with the advective transport.( 3) It has been shown that far~field transport of dissolved constituents in rivers can be satisfactorily treated by a two-dimensional model in which vertical variations of velocity and concentration are neglected.(l,4,5) Such a model, however, retains transverse variations of river-bottom topography and velocity.

Consider a section of a steady natural stream as shown in Figure 2.1. The origin of the coordinate system is placed on the near shore. The x-axis is taken positive in the downstream direction, the z-axis is directed verti-cally downward from the water surface, and the y-axis is directed across the stream. The steady-state mass balance equation for a vertically mixed radio-nuclide concentration may be written(l) as follows:

2.1

x • Downstream

y • Across Stre•m z • Vertically Downward

From W•ter SurfKa

u~y) •Velocity in x Direction Varying with y

z

FIGURE 2.1. Model of an Infinitesimal Stream-Tube in a Natural Stream (Redrawn from Yotsukura and Cobb)(l)

with boundary conditions

where

aC ay = 0 at y = O and y = B

C = radionuclide concentration, curies/ft3; d = stream depth, ft;

E = lateral turbulent dispersion coefficient, ft2/sec; y u = stream velocity, ft/sec; and A= decay coefficient and= (ln 2)/half-life, sec-1•

{2 .1)

Because, for a real stream, u and d will be functions of the transverse coordinate y, Equation (2.1) will generally not have a closed-form analytical solution. A more tractable from of the equation is obtained through introduc-tion of a new independent variable q (stream-tube coordinates), defined by

q = JY {ud)dy 0

2.2

(2 .2)

The quantity q, ft3/sec, is the cumulative discharge measured from the near

shore as represented in Figure 2.I. Hence, as y + 8, q ? ~' where B is the· river width, ft, and Q is the total river flow, ft3/sec.

Substitution of Equation (2.2) into Equation (2.1) yields the fpllowing transport equation:

(2 .3)

In the decay term, the velocity u may be replaced by the sectional mean value

u. If this is done, the decay term may be removed through the transformation

- AX/u C(x, q) = x(x, q)e

The result is the following transport equation for the nondecaying concentration:

:~ = a~ [(Eyui) :~]

(2 .4)

(2 .5)

The quantity E ui is known as the "dispersion factor." Yotsukura and Sayre(ll) haveyshown that the variable dispersion factor may be replaced by

a constant factor Eyud2, ft5/sec2, where

is the discharge-weighted mean value. The value of this coefficient may be

computed by the method presented in Section 2.2. Equation (2.5) may now be written

(2 .6)

2.3

where

D = Eyud2 = constant dispersion factor

Equation (2.6) is a standard diffusion equation which has a closed-form ana-lytical solution.

Assume a steady point-source discharge emitting a constant W ci/sec is located at the point x = 0, y = Ys· Because there is a one-to-one correspon-dence between the transverse distance y and the cumulative discharge q, the point-source discharge may be located at x = 0, q = qs. A closed-form solu-tion to Equation (2.6) which satisfies the condition that there be no flux of material across the bounding surfaces is given by

( 2 2 ) n1Tq ] exp - n ~2 Dx cos~ cos nQg (2 .7)

If the liquid effluent is injected as a line source perpendicular to the river flow, the solution may be obtained by integration of Equation (2.7) over the source dimensions. If the source is located in the river between distances

Ysl and ys2 (cumulative discharges qsl and qs2), the line-source solution may be obtained from Equation (2.7) by integration with respect to qs between the

limits qsl ~ qs ~ qs2

where

1T a= 2

xline(l) = ~[l + 2 exp[- n2ir2ox]sin na Q2 na •

(2 .8)

2.4

2.1.3. Model Applications

It is useful to write Equatlons (2. 7} and {2 ~8) -;n---dimension less form

-n2,lx e cos nuqs cos nuq (2 .9)

x L ine(l) = 1 + 2 (2 .10)

where

q = q/Q is the dimensionless cumulative discharge; - w x = xlq is the dimensionless concentration relative to the fully

mixed value; and

x = D~ is the dimensionless downstream distance. Q

The dimensionless form of Equation (2.9) is illustrated in Figure 2.2, which shows near- and far-shore concentratigns resulting from a near-shore point discharge. For a given downstream location and given flow parameters, the dimensionless concentration for either shoreline may be obtained directly from the two curves. The near-shore concentration exhibits the expected x1' 2 dependence for two-dimensional mixing until the influence of the far shore is felt. Both curves in Figure 2.2 approach unity (complete sectional mixing} for large values of x. Hence, for a given set of flow parameters, the downstream distance to sectional homogeneity (mixing distance) can be esti-mated directly. (Note that the mixing distance for a shoreline discharge is four times the mixing distance for a centerline discharge.)

Application of the model requires determination of stream channel geome-try, the cross-stream distribution of flow, and the dispersion factor at rep-resentative river cross sections downstream of the effluent discharge.

2.5

><

z Q I-

The preferred method of determining the flow cross--sectional distribution is by current meter measurements using standard stream-gaging techniques. Because it is not always practical to obtain velocity measurement at every river cross section at which concentration distributions are desired, trans-verse velocity distributions may be estimated from observed stream-bottom profiles and the application of steady-state flow equations such as Manning's formula to channels of complex cross section.(6)

Fischer(l) has shown that the lateral mixing coefficient is increased in a bending stream, varying inversely as the square of the radius of curva-ture. In general, to obtain realistic transport estimates, values of the lateral mixing coefficient are best determined by onsite tracer studies. Equations (2.9) and (2.10) may be modified as follows to account for a disper-sion factor that varies in the direction of flow:

i x Ox = D(x) dx 0

(2.11)

If the dispersion factor is known for each river cross section of concern, the integral can be evaluated by simple numerical integration.

The computer program presented here takes the· longitudinal variation of the dispersion factor into account.

The stream-flow distribution and dispersion factors for use with the dispersion model may be calculated using program TUBE9 covered in Section 2.2.

2.1.4. Program STTUBE - User Documentation

2.1.4.1 Program Description

(1) Restrictions

Program STTUBE is written in both FORTRAN and BASIC languages. The

program is used for dispersion calculation in a river with the following restrictions:

2.7

(a) Released substance is steady and nondecaying.

(b) River is not stratified; pollutant is assumed to be vertically well mixed.

(c) River flow is steady.

(d) River discharge is constant over entire portion of the river.

(e) Release of contaminant is from a point source or a horizontal line source perpendicular to shore.

(f) Velocity field and dispersion factors must be known or estimated at each cross section to facilitate computations of stream-tube coordinates and the dispersion factor. This is usually accomplished by first running program TUBE, Section 2.2.

(2) Computation Procedure:

Equations (2.9) (point source) and (2.10) (line source) are used for computation. The major steps are:

(a) Read input data.

(b) Calculate travel time~

(c) Calculate at each segment normalized concentration according to whether a point or line source is chosen.

(d) Repeat (b) and (c) until all segments are calculated.

(e) Print results. The computer program listings and flow chart are included in Appendix A.

2.1.4.2 Input Requirements

Data required for executing the program are as follows:

2.8

Variable

Q

Q3 Ql

Q2

Description Total river discharge Plant discharge Cumulative discharge coordinate at

beginning of line source Cumulative discharge coordinate at end

of line source Xl Number of cross sections Y2 Number of stream tubes x D

u

Downstream distance of each cross section Dispersion factor for each segment Cross-sectional average velocity for each

segment

NOTE: For point source, use Q1 = Q2

Unit cf s cf s cf s

cf s

ft ft5/sec2

ft/sec

The location of each point across the river is given in terms of the cumulative discharge coordinate q/Q. The location in terms of linear distance from the near shore is found from the table of stream-tube coordinates versus q/Q, which may be determined by program TUBE in Section 2.2 or by other means.

2.1.4.3 Program Operation

Program STTUBE is an interactive code. All inputs or outputs are on the terminal. If the FORTRAN version is used, all physical data should be entered with an {8Fl0.0) format, including the decimal point. After the program is properly "logged in" to the computer and loaded, the program will prompt the user for input data as follows:

(1) ENTER TOTAL RIVER DISCHARGE, CFS

> River discharge Q in cubic feet/sec is entered.

(2) ENTER PLANT DISCHARGE, CFS

> Discharge Q3 from plant in cubic feet/sec is entered.

(3) ENTER CUM. DIS. COORDINATE Ql AT BEG. OF LINE SOURCE, CFS

> 2.9

(4) ENTER CUM. DIS. COORDINATE Q2 AT END OF LINE SOURCE, CFS r

FOR A POINT SOURCE, Ql = Q2 > For a point source, enter Q2 the same value as Ql. For a shore-line point source, Ql = Q2 = 0. This is simply the line source defined in cumulative flow rate coordinates as discussed above.

(5) ENTER NO. OF X-SEC

> The river is broken into a number (up to 10) of longitudinal seg-ments where dispersion factor and velocity are defined. The spacing between cross sections is arbitary and does not have to be constant. Generally, the spacing between cross sections should be long where the river has a uniform cross section, and correspondingly shorter where the cross section is more varied.

(6) ENTER NO. OF STREAM TUBES

> This number is simply the number of divisions of the river cross sections. Each division carries an equal fraction of the total flow rate of the river. This number is used to specify points across the river for which the computation of concentration is performed; e.g., for four stream tubes, a computation would be performed at q/Q = 0, .25, .5, · .75, and 1.0. This number has no effect on the accuracy of the computation and only affects the number of points printed out across the river.

(7) ENTER FOR EACH X-SECTION, START! NG AT SOURCE PLANE, .THE DOWNSTREAM DISTANCE -X, FT

THE DISPERSION FACTOR - D, FT**5/sec**2 THE CROSS-SECTIONAL AVG. VELOCITY -U, FT/SEC Enter in a line the requested information, separated by commas; e.g., 5544, 36.6, .54 means 5544 ft down stream, a dispersion factor of 36.6 ft5/sec2, and an average velocity Q/A of .54 ft/sec. (In FORTRAN version, enter values in FlO.O format, including decimal point.)

2.10

2.1.4.4 Output Description

For each cross section, the program prints out the downstream distancP.,

the travel time to that segment, concentrations normalized to the average concentration, and dilution factors at points across the river.

2.1.5. Sample Problem

A hypothetical river is used to show the operation of the program. How-

ever, reasonable values of the necessary parameters are used. A small blow-down stream of 10 ft3/sec is discharged through a submerged pipe into the river. The average flow in the river is 4800 ft3/sec in the region of

interest.

Representative cross sections and other data were available in this case from a previous unsteady flow analysis of river flooding. Equation (2.21) from Section 2.2 was used to estimate conservatively small values of the dispersion factor at each cross section:

-2 1/2 R 1716 -0 = au g h n

1.49

where

u = the average stream velocity estimated to be Q/A, (ft/sec); Rh = the hydraulic radius, ft; and -n = the average Manning coefficient.

The factor a is estimated by the conservative relationship, Equation {2.22), in Section 2.2:

subject to

[l .49 B Rh 1/612

a = o.056 "Yg n RC

a > 0.23

2.11

where B is the river width, ft, and Rc is the radius of curvature, ft. A representative Manning coefficient of n = 0.035 and a constant flow rate of 4,800 cfs were chosen for the calculations. The computation of the parameters for the problem are illustrated in Table 2.1.

TABLE 2.1. Computation of Parameters for Stream-Tube Model Example

x A Rh u D ft ft2 ft ft/sec ft5sec-2

0 7,270 16.99 0.66 40.8

5,544 8,826 18.84 0.54 36.6

11,086 9,707 21.78 0.50 47.3

16,632 10,388 24.93 0.46 58.7

22,176 13,170 24.35 0.36 33.7

27,720 15,953 23.83 0.30 22.0

33,264 14,259 25.00 0.34 32.3

38 ,808 12,565 26.19 0.38 46.l

44,352 14,183 26.64 0.34 38.7

49,896 15,800 27.14 0.30 31.8

To estimate the size of the line source, it would have been possible to use a jet model to simulate the shape of the jet close to the discharge. Then it would be jet at some dissipated.

possible to estimate the volume of river water entrained in the point downstream where most of the jet momentum would have been This procedure is illustrated in Figure 2.3.

The cumulative flows in Figure 2.3 are defined as follows: q1 is the cumulative river flow between the shoreline and the near side of plume

q2 = q1 + jet discharge + water entrained from river at x = x1 In this example, the procedure was not followed because of the small size of the discharge. Instead, it was assumed that the source was approximately a point source, located at q1 = q2 = 10 cfs, which is only a short distance in flow coordinates from the near shore because the total river flow Q is 4,800 cfs.

2.12

y DISTANCE ACROSS RIVER

RIVER FLOW

PLUME FROM JET MODEL

APPROXIMATE LINE SOURCE

x - DISTANCE DOWNSTREAM

FIGURE 2.3. Jet Discharge Approximated by a Line Source

The computer run is shown in Figure 2.4. Normal concentrations and dilu-tions are printed for points across the river at each of nine cross sections. Concentrations are plotted in Figure 2.5 for the near shore (q = O), far shore (q = 4800 cfs), and the middle of the flow (q = 2400 cfs) as a function of distance downstream. A check of the validity of the model is that all normal-ized concentrations converge to the value of 1.0.

2.1.6 Model Confirmation

The transverse mixing characteristics of the Missouri River in the vicin-ity of the Cooper Nuclear Station near Brownville, Nebraska, were investigated using the fluorescent-dye tracer technique.( 14 ) Rhodamine WT dye, introduced continuously into the plant's once-through circulating water system, was used to simulate the waste heat. Transverse profiles of dye concentration, depth, and velocity were obtained at several cross sections in the 6-mi reach immedi-ately downstream from the plant. The results of the concentration measurements

are used in the model confirmation.

2.13

READY RUN STTUBE COMPUTATION OF TRAVEL TIME AND DILUTION IN A RIVER IN STREAM TUBE COORDINATES.

ENTER TOTAL RIVER DISCHARGEv CFS ? 4800 ENTER PLANT DISCHARGE, CFS ? 10 ENTER CUM. DIS. COORDINATE Qi AT BEG+ OF LINE SOURCE' CFS ? 10 ENTER CUM. DIS. COORDINATE Q2 AT END OF LINE SOURCE, CFS FO~ A POINT SOURCE, Q1=Q2

? 10 ENTER NO+ OF X-SECTION. ? 10 ENTER NO. OF STREAM TUBES ? 10 ENTER FOR EACH X-SECTION, STARTING AT SOURCE PLANE

THE DOWNSTREAM DISTANCE - x, FT THE DISPERSION FACTOR - Dv FT~5/SEC~2 THE CROSS-SECTIONAL AVG. VELOCITY - Uv FT/SEC

X-SECTION 1 ? Ov40.8v.66 X-SECTION 2 ? 5544,36.6v+54 X-SECTION 3 ? 11086v47.3,.5o X-SECTION 4 ? 16632v58~7,.46 X-SECTION 5 ? 22176,33+7?+36 X-SECTION 6 ? 27720,22.0,.3 X-SECTION 7 ? 33264,32.3,.34 X-SECTION 8 ? 38808,46+19+38 X-SECTION 9 ? 44352,39.7,.34 X-SECTION 10 ? 49896,31+8,.3 THANK YOU

FIGURE 2.4. Sample Problem - Program STTUBE (BASIC version) (Note: In FORTRAN version, values entered in FlO.O format, including decimal point.)

2.14

FIGURE 2 .4. {contd)

CONCENTRATIONS NORMALIZED FOR X EQUALS 5544 TRAVEL CUM.DIS.FRACT.NORM CONC

0 5.84586 +1 4.46975 +2 1+99795 +3 .4

i::-. ..., .6 +7 .8 .9 1

+522102 • 07976'19 7.12359E-03 3+71754E-04 1 • 12653E ·-O::'i 0 0 0

TO COMPLETELY ~IXED CONC+ TIME EQU~LS 2+56667

DILUTION 82+1093 107. :389 240.246 919.36 6017+91 67381.8 1.29118Et06 4.26088Et07 1.00000Et20 1.00000Et20 1+00000Et20

FOR X EQUALS 11086 TRAVEL TIME EQUALS 5.52714 CUM.DIS.FRACT.NORM CONC DILUTION

0 4.05013 118.515 • 1 .2 .3 .4 +5 .6 +7 +8 +9 1

3+56055 2.41915 1. 270~~ +515519 +161689 .0391929 7 +34252E-.. 03 1+06269£ .. -0:5 1.19388£--04 2+02656E-05

134.81 198.417 377.865 93l .101 2968 +.67 12247.1 65372.7 451683 4+02050Et06 2+36855Et07

FOR X EQUALS 16632 TRAVEL TIME EQUALS 8.73663 CUM+DIS.FRACT.NORM CONC DILUTION

0 3+14593 152+578 .1 2+91066 164.911 .2 +3 .4

i::· +>:I

.6 • 7 .8 .9 1

2+30525 1. 56~?9 .90704 .450617 +191635 + 0697691 . .0217832 6.05834£-03 2+64853E-03

2.15

208.22 307+122 529.l94 1065+21 2504.76 6879.83 22035.3 79229.7 1012:~2

FIGURE 2.4. (contd)

FOR X EQUALS 22176 TRAVEL TIME EQUALS 12.4927 CUM.DIS.FRACT.NORM CONC DILUTION

0 2.71196 176.994 + l

') t A".. . ;:~ .4

a::· .... J

+6 .7 .!3 .9 :t

2. 5~i975 2. l ~i247 :L +I.> :I. 2~.'i2 :1..076)21. .63991.6 .339007 .160127 • 0679196 .o:.;!769:1.7 .Ol.68132

if:P.518 2;;!2 + 999 297.671 446.00H

. 7~i0. 098 l415.9 2997.62 7067.18 :I. 7333. 7 ~?8!'.'i49

FOR X EQUALS 27720 TRAVEL TIME EQUALS 17.1594 CUM.DIS.FRACT.NORM CONC DILUTION

0 2+5236l 190.203 .1 2+40049 199.959

'") . ,,.._

.6 ., . ;

.B

.9 1

2.066 1 • .1>0885 1 • :I. 33!59 .7227:1.6 .417003 .2Hl107 .104c>2:L .049836 • ()339429

298.349 423.432 66>4 .162 11t"'i1.07 22()(). 76 4~iff7 + 97 9631.5B l.4141.4

FOR X EQUALS 33264 TRAVEL TIME EQUALS 21.9719 CUM.DIS.FRACT.NORM CONC DILUTION

0 2+37327 202.252 • :L

'") + 11.'.,

.3 +4 f !)

+6 .7 .8 .9 1.

2. 270!'.'i7 :l • 9B83f.l :I. • !'.)9;383 :I .• :J.6942 • 7B!'.'i44:1. • 4f.l31!56 +272968 • :I. 4:39!'.'i4 .0771846 • 0569:1.()4

2.16

24:1. + 402 30:1 .• :t.61 4:1.0.462 6:1. :t .• 121 993.469 :l 7!'.)8. 4t"'i ~~3:~4 + 4 62U3•86 94;34. ;31

FIGURE 2.4. (contd)

FOR X EQUALS 38808 TRAVEL TIME EQUALS 26.2497 CUM.DIS.FRACT.NORM CONC DILUTION

0 2+197 218.48 • l

'') f A*.o

+3 .4

c;· • ~J

+6 "1 . ,

+8 .

c CMIXED

6.0

5.0

4.0

3.0

2.0

1.0

10,000

CENTER OF FLOW q = .5

FAR SHORE q = 1

20,000 30,000 40,000 50,000 60,000 70,000

x- DISTANCE DOWNSTREAM- FT

FIGURE 2.5. Normalized Concentration at 3 Points Across River Example

2.1.6.1 Model Parameters

The application of the model requires river characteristics and dye release rate. Only the results of those portions of the river where measure-ments were available are compared. These are transects 3, 5, 7, 9, and 10 at miles 531.5, 530.5, 529.0, 527.0, and 526.11. The Manning n's are calculated as 0.018 using average velocity u = 6.5 ft/sec, average depth = 13 ft, and slope= 0.00019. The radii of curvature at the Langdon and Aspinwall bends are 6400 ft and 3400 ft, respectively as cited. The radius of curvature at the lower Brownville Bend is estimated at 8882 ft.

2.18

Lateral dispersion factors required are obtained from the experimental data and from the estimation by Program TUBE (Section 2.2). Table 2.2 shows the··rtts~ersionfactors-es·t+mat·etl-·by····the--thr·ee .. d.t.fferent--methodsJn .. .Pr_o.gr.am_IUBL

and the dispersion factors calculated for Transects 3, 5, 7, and 9 from the transverse mixing. coefficients determined by the simulation method.{l4) The simulation method is based on the steady-state convection-diffusion equation in a meandering-coordinate system(16 ) similar to the one employed by Yotsukura et al.(12 ) In addition, a dispersion factor of 13,600 ft5/sec2, as determined by method of moments,(l4) is also used for the entire test reach.

The longitudinal dispersion coefficient Ex is estimated from Equa-tion (2.40) as 22.5 ft2/sec. The river flow rate on the test day was 56,100 cfs. The plant discharge was 1455 cfs. The discharge canal was located at the right bank of the Missouri River. The dye concentration was nearly uniform in the canal, and the average concentration of 35.6 ppb was close to the theoretical concentration of 36.8 ppb. The canal discharged into the river as a point source and resulted in a fully mixed concentration of 0.95 ppb in the river. The constant rate of discharge of the dye was main-tained from 0800 hours until 1333 hours.

2.1.6.2 Model Results and Comparisons

Program STTUBE is run with the above dispersion factors for comparing with field measurements. Figures 2.6a and 2.6b show the distributions of con-centration calculated with dispersion factors obtained from Program TUBE. It appears that Method 2 gives the best fit to the experimental data, followed by Methods 1 and 3. The comparisons among the results obtained from variable lateral dispersion factors by the simulation method (where available), a con-stant dispersion factor of 13,600 ft5/sec2 for the entire test reach, and the experimental data are shown in Figures 2.7a and 2.7b. Both constant and vari-able dispersion factors show reasonably good fit. Notice, however, that except for Method 3 in Program TUBE, all dispersion factors give better fit with the experimental data the further downstream the transect is located.

2.19

ll ~ 12

10

8

6

4

..0' 2 0.. 0..

z z 0 I- 0

6

5

4

TRANSECT 7 I MILE 529.0 j

-·0·-DISPERSION FACTOR BY METHOD 1 f~ ·

3

2

.... e::, .... DIS PERS I ON FACTOR BY METHOD 2 ~ --o--DISPERS ION FACTOR BY METHOD 3 A_.· .... -+-MEASURED DATA • .zt' a···

-&. 4 TRANSECT 9 c.. :z MILE 527.0 :z 3 0

!;;( 2 0::

!Z ~ 1 :z 0 u 0

4

3 TRANSECT 10 MILE 526, ll

2

I

0

1.0 0.8

.~··

0.6 0.4 0.2

NORMALIZED CUMULATIVE DISCHARGE, qc/QR

0

FIGURE 2.6b. Distribution of Dye Concentration with Respect to Normalized Cumulative Discharge, Transects 7, 9, and 10

2.21

8

TRANSECT 3 MILE 531.5

6 --er-VARIABLE Ey USING SIMULATION

METHOD BY SAYREI5

4 ····6···· CONSTANT DISPERSION FACTOR 03, 600 tt5/sec2)

~MEASURED DATA .c

2 c.. c.. z z 0 I-

4 TRANSECT 7 - -(r - VARIABLE Ey USING SIMULATION

3 MrLE 529.0 METHOD BY SAYRE 15 • ····~····• CONSTAlllT DISPERSION FACTOR 2 m. 600 115/sec21 -+-- MEASURED DATA l

0

4 ..0 ·TRANSECT 9 8: l MILE 527.0 z z 2 0 ~ ~ 1 I-z LI.I (.)

0 z 0 (.)

4 TRANSECT 10

3 MILE 526.11

2

1

0

1.0 0.8 0.6 0.4 0.2 0

NORMALIZED CUMULATIVE DISCHARGE, q/OR

FIGURE 2.7b. Distribution of Dye Concentration with Respect to Normalized Cumulative Discharge, Transects 7, 9, and 10

2.23

TABLE 2 .2. Dispersion Factors (x lo3), ft5/sec2, by Program TUBE

Transect Method 1 Method 2 Method 3 Method (

bl Simulation 14 > 1 1.9839 1.5954 1.0915 N/A 3 3.2454 4.2456 , 1.0743 6.016 5 2 .5900 6.1992 1.0240 16.656 7 1.4739 6.1414 1.07.65 7.388 9 24 .775 42.565 . 4 .0028 50.721 10 1.5015 8.0017 0.7082 N/A

2.2 PROGRAM TUBE - ESTIMATION OF THE STREAM-TUBE LOCATIONS AND THE DISPERSION FACTORS

2.2.1 Introduction

This program is used to estimate the locations of the stream tubes, and the disperision factors for use by the program STTUBE. A stream tube is a p.ortion of the cross section of a river carrying a fixed quantity of the flow. Program STTUBE requires that the river be divided into N stream tubes, each carrying (1/N)th of the total flow.

If there are detailed velocity measurements across the river, the width of each stream tube (carrying equal flow) may be found directly. If there are no detailed measurements, program TUBE may be used to estimate the. flow, stream-tube locations, and dispersion factor at each cross section based on the bathymetry and friction coefficient.

2.2.2 Theory

2.2.2.1 Velocity Distribution

In a wide, relatively shallow, open~channel flow, the steady-state flow rate may be expressed by the Chezy formula:

u2 C2d =-gs h

(2.12)

2.24

where

where

and

d = depth, ft;

Ch = Chezy coeffiGient; n =Manning coefficient; u = velocity in the direction of flow, ft/sec; s = energy slope; and g =acceleration of gravity, ft/sec2•

The velocity in the direction of flow at any point across the river is

u = f i13

n

f = --.,,...--..,,;.Q..,..,..... __ IB !!5/3 dy o n

Q =river flowrate, ft3/sec; and B = river width, ft.

(2.13)

The cumulative flowrate q at any point across the river is

l y d5/3 dy Q o n-

q - ___,,,..--"'='""""'..----- IB !!5/3 dy o n

(2 .14)

Therefore, the cumulative flow rate q is now known as a function of the distance y across the river and the Manning's coefficient. Equally spaced values of q can, therefore, be determined to define the width of each stream tube.

2.25

The validity of the method for velocity distributions based on the Man-ning formula is supported by a study performed by Sium,(lO) which investi-gated the relationship between vela.city distribution and depth in a large number of cross sections in a major river. The results are summarized in Figure 2.8. The straight dotted line represents Manning's equation with constant lateral friction coefficients. Agreement is reasonably good, and would probably improve if some estimate of the lateral variation of Manning's coefficient could be made.

2.2.2.2 Dispersion Factor

Equation (2.5) in Section 2.1.2 describes the correct usage of the dispersion factor

~ = 2. [tE uci2) ~] ax aq \ 1 y aq where

EY = lateral dispersion coefficient at distance y across river, ft2/sec;

u = the local stream velocity at point y, ft/sec; and

d = the depth at point y, ft.

(2.15)

The general case of Equation (2.15) does not have an analytical solution. A simplification of Equation (2.15) that is tractable involves removing the term Eyud2 from the derivative

( 2 .16)

The term Eyud2 is called the Dispersion Factor, D.

2.26

POINT FLOWRATE RELATIVE

5

2

TO WIDTH - 1.0 AVERAGED FLOWRATE

q

'fi 0.5

0.2

0.1--~------~--_..~------...-~~~~--~~--~__. 0.2 0.5 1.0 2

h -=- DEPTH RELATIVE TO WIDTH-AVERAGED DEPTH h

FIGURE 2.8. Correlation of Channel Depth Versus Flow Data (after Yotsukura and Sayre, Reference 11)

2.27

4

Three approximations for computing the dispersion factor are presented here. All three give about equal values for rectangular shaped channels, but differ for irregularly shaped channels. Dispersion factors by all three methods are calculated by program TUBE. These calculations presume that detailed velocity measurements or dye dispersion studies are not available.

(1) METHOD 1

The dispersion factor in this method is taken as the flow-weighted mean of Eyud2 across the channel

where

and

Ey

u*

8

g

2 1 IQ 2 D = E ud = Q E ud dq y .. 0 y

= eu*d (lateral dispersion coefficient)(13);

= ut9' = shear velocity; h

=constant coefficient defined in.Section 2.2.2.3.2; = acceleration of gravity;

ch = Chezy coefficient, Ch

n = Manning coefficient. n

Combining all terms,

rQ 2 D = !_ Jn au Yg' n i7 /6 dq

Q 0 1.49

2.28

(2 .17)

( 2 .18)

{2) METHOD 2

In this method, the dispersion factor is based cient EY calculated from the bulk properties of the charge weighted average of ud2

where

r -2 -1 IQ 2 D = • ud = EY Q . ud dq y . 0

- Q u = A;

A = cross sectional area, ft2; and

Rh =hydraulic radius of the channel, ft.

All barred quantities are channel averaged.

(3) METHOD 3

on the dispersion coeffi-channel, and the dis-

{2.19)

The third method uses only channel averaged values

. (2 .20)

Combining all terms,

2.29

2 o =a Q2 g112 R~7 ' 6 n/1.49

A

2.2.2.3 Discussion

(1) Choice of Methods

(2.21)

For irregularly shaped channels, each of the three methods gives differ-ent dispersion factors. Method 1, even though it is more complicated, is not necessarily more correct than the other methods in calculating the dispersion factor for the stream tube model. Yotsukura and Cobb(l) and Jackman and Yotsukura( 2) have used methods 2 and 3 with reasonable results.

The lower values of dispersion factor produced generally are conserva-tive. However, the user is cautioned that small dispersion factors are not always more conservative than larger ones. There are cases where the opposite is true; e.g., when the discharge is on the near shore and the intake is on the far shore. In these cases, however, it would be conservative to assume complete mixing across the river.

If the bulk properties of the river channel (i.e., Rh, A) are already known, the dispersion factor may be conservatively estimated from Equa-tion (2.21) directly. This was the procedure used in the example in the text.

(2) The Factor a

This factor is an emperical coefficient used in the expressions for lat-eral dispersion. This factor is usually taken to be conservatively 0.23, which is a value considered to be representative of straight natural channels. The value of a can increase significantly in curved channels. For instance, Yotsukura et al.(12 ) reported a varying from 0.6 in gradually curved channels to 10 for sharply bending channels in the Missouri River. Fischer(?) has

found average a values ranging between 0.5 and 2.5 in curved channels. Although no rigorous theory exists for this phenomenon, a seems to be related to the reciprocal of the square of the radius of curvature of the channel.

An empirical, conservative relationship for a is used in the computer program

2.30

- (1.49 B Rh 1/6) 2 a - • 056 -vg n R

c

subject to a > o.23

where B =width of the channel, ft;

n =representative average Manning's coefficient; Rc =radius of curvature of the centerline of the channel, ft; and

the other terms are as previously defined.

(2.22)

This computation for a is conservatively small because it is based on laboratory flume data of Fischer,(?) which underpredicts those values that would be expected in real rivers, as shown in Figure 2.9. The lower value of a is limited to 0.23 because this is the widely reported value typical of straight natural river channels (RC~ oo).

(3) Manning's Coefficient n

In simple channels, the roughness along the wetted perimeter may be dis-tinctly d{fferent from place to place, but the mean velocity can still be computed by a uniform-flow formula without actually subdividing the section.

In applying the Manning formula to such channels, it is sometimes necessary to compute an equivalent n for the entire perimeter and use this value for the computation of the flow in the whole section.

For the determination of the equivalent roughness, the water area is

divided into N parts, of which the wetted perimeters P1, P2, •.• , PN and the coefficients of roughness n1, n2, •.• , nN, are known. Assuming that the total force resisting the flow is equal to the sum of the forces resisting the flow developed in the subdivided areas, the equivalent roughness coefficient is(9)

2.31

10

MISSOURI RIVER

5

2 •

2 LABORATORY FLUME

1.0

0.5

0.2

0.1--~~~--~~---___...__ .......... __ ......._ .................. ~~~~_._~__..__ ........ 1 2 5

1.49 B RH 116

v'9 n Re

10 20

FIGURE 2.9. Correlation of Factor a (after Yotsukura and Sayre, Reference 11)

2.32

40

Z, DEPTH FT

5

10

15

1

0

LEFT BANK

100 200 300 400 500 y- FT

APPROXIMATE RIVER BOTTOM

600 700

FIGURE 2.10. Approximate River Cross Section

1/2

where P is the total wetted perimeter.

2.2.3 Computation of Cumulative Flow and Dispersion Factor

2.2.3~1 Cumulative Flow

6

800 900

RIGHT BANK

( 2 • 23)

Referring to Figure 2.10, the river cross section is represented by an arbitary number (M-1) of straight-line segments in the y-z plane. The first point on the near shore and the last point on the far shore are the points where the water surface intercept the shoreline. River flow is calculated in

each of the irregular trapezoidal river segments defined by the straight-line approximations.

2.33

j - 5/3 L: Yi ( Ay) i

cumulative i=l n. qj = Q M-l

1

where

flow

I: - 5/3 y. 1 (Ay) i i=l n. 1

Yi = the average depth in the segment,

= (yi + Yi+l)/2, ft; Ay.

1 = width of segment i = Yi+l -yi ft; and

ni =average Manning's coefficient in segment i.

(2.24)

Equation (2.24) defines the cumulative flowrate at points across the river, as depicted in Figure 2.11. Linear interpolation of the table of cumulative q versus y yields the value of width of each stream tube.

RIVER CROSS-SECTIONAL DISTANCE, y FT

FIGURE 2.11. Cumulative Flow Computed by Eq. (2.24) ·

2.34

2.2.3.2 Dispersion Factor

(1) Method 1

The Dispersion Factor in Equation (2.18) is evaluated from the expression

where D = dispersion factor, ft5/sec2; Q = the total river flow, ft3/sec;

6q. = incremental flow in segment i, ft3/sec; l

= qi+l - qi; 6Yi =width of segment i, ft;

= Yi+l - yi; and ni = Manning 1s coefficient in segment i.

(2) Method 2

(2 .25)

1 (Q 2 The terms in Equation (2.19) D = EY Q J0 ud dq, are evaluated as

fol lows:

The term

is approximated as

1 M-1 ( )2-~ 6q. d. - L.J l l Q i=l 6y i

. 2 .35

2 The dispersion coefficient Ey, ft /sec, is based on the bulk properties of the channel

where

where

Rh= the hydraulic radius= A/P, ft; A= the cross-sectional area, ft2, approximated as

M-1 A = L d. ( AY,· ) ;

. 1 l l=

P = the wetted perimeter, ft, approximated as

u* = shear velocity = uY9 n 1/6 1.49 Rh

, ft/sec

u = the average stream velocity Q/A, ft/sec; and -n =the representative average Manning's coefficient.

rlence,

2.36

{2.26)

{2 .27)

(2 .28)

(2 .29)

(3) Method 3

The dispersion factor, D, in Equation (2.21), is calculated only from the bulk properties of the channel segment

a g_2gl/2 R 17/6 n A · h D = ---.....,,....,.-----1.49

2.2.4 Program TUBE- User Documentation

2.2.4.1 Program Description

(2 .30)

Program TUBE is written in both FORTRAN and BASIC languages. The maximum number of points defining the cross section of a river segment is 21.

Equation (2.24) is used to calculate cumulative flow ·qj. The value of the width of each stream tube is computed by linear interpolation of the table of q versus y. The factor a is calculated with the conservative relationship Equation (2.22). Equations (2.25), (2.29), and (2.30) are used to calculate dispersion coefficients D, corresponding to Method 1, 2, and 3.

The major steps of the computation are (a) Read input data. (b) Calculate cumulative flow. (c) Interp6late q versus y table for width of stream tube. (d) Compute D with Method 1. (e) Compute D with Method 2. (f) Compute D with Method 3. (g) Return to step (a) until all segments are calculated.

Results are output as computed. The listings of computer programs and flow chart are included in Appendix B.

2.2.4.2 Input Requirements

Input data required are as follows:

2.37

Vari able Nl Q2 N9

N

N2

y' z

R9

Description Number of stream tubes Total river flow Constant Manning's coefficient Manning's coefficient between two

consecutive points in a cross section Number of points defining a cross-section y-, z-coordinate of the points defining

cross sect ion Radius of curvature

2.2.4.3. Program Operation

Unit

cf s

ft

ft

Program TUBE is an interactive code. If the FORTRAN version is used, all numerical data should be entered, with an (8Fl0.0) format, with decimal points. After properly "logging in" to the computer and loading the program, the pro-gram will prompt the user for input data:

(1) ENTER NUMBER OF STREAM TUBES DESIRED

>

(2) INPUT TOTAL RIVER FLOW, CFS

>

(3) ENTER CONSTANT MANNING's COEFFICIENT - IF VARIABLE, ENTER 0

> Enter the Manning's coefficient - either a constant for the entire cross section or a zero {0)--which will signal for input of variable coefficients for segments between two consecutive defining points.

(4) ENTER NUMBER OF POINTS DEFINING CROSS SECTION

>

(5) ENTER Y AND Z COORDINATES OF EACH POINT DEFINING CROSS-SECTION, FT

> Enter the points in y-z plane defining the cross section.

2.38

(6) ENTER MANNING COEFFICIENT FOR EACH SEG OF THE X-SECTION BETWEEN 2 CONSECUTIVE POINTS

>

This question will appear only if the answer to (3) is zero (0). Enter the Manning coefficients for the portion of the channel between two consecutive defining points.

(7) INPUT RADIUS OF CURVATURE, FT

> If the segment of river in question is straight, enter a large

6 number, say, 10 •

(8) Program returns to step (4) for a new cross section. To leave the program at this point, type in any n.egative number.

2.2.4.4. Output Description

For each cross section, the program prints out the table of the loca-tion of the boundaries of each stream tube, the hydraulic radius, the cross-sectional area, the factor, a, (>0.23), and the dispersion factors, D, ft5/sec2 , computed by methods 1,-2, and 3.

2.2.5 Sample Problems

2.2.5.1 Example Problem 1

The cumulative flow and dispersion factors for a river section will be computed. The river cross section in question is shown in Figure 2.10. It is represented by 7 points, which define 6 segments:

Geometr~ Friction Point y-ft d-ft Segment a Manning Coefficient

' 1 0 0 1-2 0.08 2 100 4 2-3 0.05 3 175 12 3-4 0.035 4 400 14 4-5 0.035 5 475 9 5-6 0.04 6 800 8 6-7 0.06 7 900 0

2.39

Flow in the river is 4500 cfs. The radius of curvature at this section is 2000 ft. The test run is shown in Figure 2.12 and is self-explanatory.

2.2.5.2 Example Problem 2

This is the same as problem 1, but the Manning's coefficient is chosen to be a constant n = 0.035. In this case, the dispersion factor calculated by method 3 is D = 26.13, as compared to D = 71.08 calculated from the more com-plete formula, Method 1. Method 2 gave an intermediate result of 47.4. A smaller dispersion'factor would predict slower dispersion. The user is reminded that smaller dispersion factors are not always more conservative than larger dispersion factors. The test run is shown in Figure 2.13.

2.3 PROGRAM RIVLAK - CONCENTRATION IN A RIVER OR NEAR-SHORE REGION OF A LARGE LAKE FROM A NONSTEADY SOURCE

2.3.1 Introduction

In many cases, routine releases of radioactive effluents are batched and infrequent, rather than continuous. In such cases, it may be important to calculate concentrations as a function of both time and space. Program RIVLAK solves for the vertically integrated concentration in a river or large lake under the following limiting assumptions: (1) Constant depth d; (2) Constant down stream or longshore velocity u; (3) Straight river channel or shoreline; (4) Constant longitudinal and lateral dispersion coefficients Ex, Ey; (5) Point discharge; and (6) Constant river width B. The discharge of radioactive material may have any release schedule, as

' determined by linear interpolation of points in a table.

This model is most useful for those cases where radioactive effluent releases cannot be considered to be continuous. Application of this model should be restricted to those portions of the river or near-shore zone of a lake removed from the influence of the discharge. Initial dilution near the point of discharge is usually controlled by momentum effects of jets, as described in Appendix A of Regulatory Guide 1.113.(8 ) When applied to the

2.40

READY RUt~ TUBE

ESTIMATION OF STREAM TUBE LOCATIONS AND DIFFUSION FACTORS

ENTER NUMBER OF STREAM TUBES DESIRED ? 10 INPUT TOTAL RIVER FLOW, CFS ? 4500 ENTE~ CONST. MANNINGS COEFF - IF VARIABLE, ENTER D .,. 0

ENTER NUMBER OF POINTS DEFINING CROSS-SECTION ? 7 ENTER Y & Z COORD. OF EACH POINT DEFINING X-SECTION, FT ? o,o ? 100,4 ? 175, 12 .,. 400, 14

? 8(1(1, 8 .,. 9(1(1, 0

ENTER MANNING COEFF FOR EACH SEG OF THE X-SECTION BETWEEN ·::. '- CONS:ECUT I VE POINTS

1 - 2 ? . 08 ·::. - ·::: ·-;:· 05 '- . •'j ·-· 4 '? . 0:35 4 - 5 ? . 0::::5 C" - 6 ? 04 ·-· . E. - 7 ? . Ub

INPUT RADIUS OF CURVATORE, FT ? 2000 THANK -,·ou

STREAM TUBE BOUNDARIES IN FT POINT NO. Y DIST FRACT OF Q

1 0 0 2

4 C"

·-' 6 7 0 ·-· 9

195. 66:3 241. E03:3 287. 60:3

379. 542 431. 2'36 4''.;18.'398 605.659

• 1 ·=· . '-·-=· . ·-·

.4 C" . ·-·

• E. 7

• I

.8 1 0 712. 32 . ·:;. 11 900 1

HYDR. RAD., FT= 8.60154 X-SECTIONAL AREA, FTA2= 7750 FACTOR BETA = .694619 DISPERSION FACTOR BY METHOD 1= 46.5925 DISPERSION FACTOR BY METHOD 2 = 38.4394 DISPERSION FACTOR BY METHOD 3 = 19.0427 ENTER NUMBER OF POINTS DEFINING CROSS-SECTION 7 -1 END OF PROGRAM TUBE

FIGURE 2.12. Sample Problem 1 - Program TUBE (BASIC version) (Note: In FORTRAN version, val uesentered in FlO.O f~rmat, with decimal ~oi~t~.) · ·

2.41

READY RUN TUBE

ES:TIMAT-ION OF STREAM TUBE LOCATIONS AND DIFFUSION FACTORS

ENTER NUMBER OF STREAM TUBES DESIRED ? 10 INPUT TOTAL RIVER FLOW, CFS ? 4500 ENTER CONST. MANNINGS COEFF - IF VARIABLE, ENTER 0 ? .035 ENTER NUMBER OF POINTS DEFINING CROSS-SECTION ? 7 ENTER Y & Z COORD. OF EACH POINT DEFINING X-SECTIO~, FT ? o-, 0 7· 1oo,4 1· 175,12 r· 400,14 "7" 475, 9 !" 800,8 ? 900,0 INPUT RADIUS OF CURVATURE, FT ? 2000 THANK YOU

STREAM TUBE BOUNDARIES: IN FT POINT ND. ·Y DIST FR ACT OF G!

1 0 0 2 186.996 • 1 ·j ·-· 236.802 .2 4 286.607 .3 5 336.413 .4 6 386.218 .5 7 444.19 .6 8 525.125 .7 ·::i 626.24 .8 10 727.356 .9 11 900 1

HYDR. RAD., FT= 8.60154 X-SECTIDNAL AREA, FT~2= 7750 FACTOR BETA = 1.30773 DISPERSION FACTOR BY METHOD 1= DISPERSION FACTOR BY METHOD 2 = DISPERSION FACTOR BY METHOD 3 = ENTER NUMBER OF POINTS DEFINING ? ...;1 END OF PROGRAM TUBE

71.0826 47.3951 26.1286

CROSS-SECTION

FIGURE 2.13. Sample Problem 2 - Program TUBE (BASIC version) (Note: In FORTRAN version, values entered in Flo.a format, with decimaJ points.)

2.42

near-shore zone of a large lake, the model presented here should be considered valid only for dispersion during the time between wind-induced current reversals.

2.3.2 Theory

All surface-water transport models contained in this section are based on the solution of the convection-diffusion equation in simple geometries, with steady unidirectional flow

ac ac ac ac E a2c + E a2C a2C -+ u -+ v -+ w-= + E -,;- >.C + W(t) - S(t) (2.31) at ax ay az x ax2 y ay2 z az~

where, u, v, w are the velocities of water in the x, y, and z directions, respectively; Ex, EY, Ez are the dispersion coefficients in the x, y, and z directions, respectively; >. is the radioactive decay coefficient; W(t) is a distributed source, and S(t) is a distributed sink.

In general, each surface-water model is developed from a simplification of Equation (2.31}. W(t) and S(t) are source and sink terms for water-solid interactions when these effects are included in the models.

2.3.2.1 River Model

For nontidal rivers, the flow is assumed to be uniform and steady. For batched releases of short duration, the dispersion in the longitudinal direc-tion may be important. In such cases, concentrations must be calculated as a function of both time and space.

Consider a section of a steady rectangular stream as shown in Figure 2.14. The mass-balance equation for a vertically mixed radionuclide concentration in a unidirectional flow field may be written as follows:

!£ + u ~ = E a2c + E a2c - >.C at ax x ax2 y ai (2.32)

2.43

FIGURE 2.14. Geometry of Simple 2-D River Model

with initial and boundary conditions

c = 0, at t = 0, c = 0, at x = + 00

aC/ ay = 0, at y = 0, y = B,

where, C = the radionuclide concentration, ci/ft3;

Ex = the longitudinal turbulent dispersion coefficient, ft2/sec; Ey = the lateral turbulent dispersion coefficient, ft2/sec; u = the flow velocity, ft/sec; and A= the decay coefficient= ln 2/half-life.

The resulting concentration at time t in a straight rectangular channel of width B and cross sectional area A, with steady downstream velocity u, corresponding to an instantaneous release at t = 0 of 1 curie of material from a vertical line source at x = 0, y = Ys is

2.44

c. = l

__ 1 __ exp [- (x - ut)2 - >..t]

... J (4 E t)A 4Ex t ~ 1T x

(2.33)

The case of a more general time-dependent release may be obtained by inte-grating Equation (2.33) with respect to time using the convolution integral

t c(t) = f c.(t-T) f (T) dT

0 l (2 .34)

where f(T) is the source release rate in curies/sec; and Ci (t - T) is the solution at time (t - T) for an instantaneous release of 1 curie at (t - T) = O.

This yields

{ [x-u(t-T)]2 }

exp - 4Ex(t - T) - >..(t - T)

exp (2 .35)

where the release rate is f(T) curies/sec. In general, Equation {2.35) must be solved by numerical quadrature.

Near the source, convergence of the Fourier series terms in Equa-tions (2.33) and (2.35) may be extremely slow. However, in this region, the effects of the far-shore are not usually important, and the. series solution may be replaced by a single image source at the near shore (see the transient lake solution, Equations (2.38) and (2.39). In this case, the solutions do not involve infinite series and present no convergence problems.

2.45

2 .3 .2 .2 Lake (Near-Shore)· Model

The lake is assumed to have a straight shoreline and to be of constant depth, with steady unidirectional flow parallel to shore. Release is postu-lated to be from a vertical line source extending from the surface to the bottom. Dispersion occurs in both the lateral and longitudinal directions with constant turbulent dispersion coefficients.

Consider a section of a large lake shown in Figure 2.15. The mass-balance equation for the vertically mixed radionuclide may be written as follows:

0

~ + u ~ = E a2c + E a2c - xc at ax x ~ y ay2

The initial and boundary conditions are

C = 0 at t = 0 C = 0 at x = + oo and at y = oo

aC/ay = 0 at y = o

( 2 . 36)

(2 .37)

The resulting concentration at point x, y and time t in a lake of depth d with steady longshore velocity u, corresponding to an instantaneous rel-ease at t = 0 of 1 curie of material from a vertical line source at x = 0, y = Ys' is

(2 .38)

The case of a more general time-dependent release may be obtained by using

the convolution integral Equation (2.34}

2.46

FIGURE 2.15. Near-Field Model

C = It f (' ) exp {- (x 4Ex u I~ = ~ j} 2 - A ( t - T 1} . 0 4'1f l["r d( t - T) '\/-x-y

-(y - y ) . -(y + y ) l ( 2) ( 2) l exp 4EY(t _s,) +exp 4EY(t _s,) di: (2 .39) where the release rate is f(i:} curies/sec. In general, Equation (2.39) must be solved using numerical quadrature.

Equations (2.38) and (2.39) are also useful for releases into rivers in the region near the source, where the effects of. the far shore are unimportant.

2.3.3. Model Application

2.3.3.1 Values of Coefficients

Application of this model requires the determination of the lateral and longitudinal turbulent dispersion coefficients Ex and EY respectively.

2.47

Dispersion coefficients should be obtained from site-specific tracer experi-ments wherever possible. However crude estimates may be obtained from the following formulae:( 14 )

where d = the average depth, ft; g = the acceleration of gravity, ft/sec2; n =the Manning's coefficient; and

ax, ay = dimensionless constants.

(2.40)

(2.41)

For straight rectangular river channels or straight shoreline, ax has a value of about 5.93 and ay is about 0.23. For curved chanels, however, both lateral and longitudinal mixing increase and must generally be determind by field observations. The a values representative of straight channels may be considered conservatively small, however. The user must be cautioned that small values of ~x and Ey are not necessarily conservative, if the concen-tration at a point downstream would be higher with greater dispersion.

Studies(l5) in the Great Lakes indicate that representative values for

EY are in the range of about 0.5 to 1.0 ft2/sec. Notice, however, that these values are typical only of the nearshore zone and not of deeper waters where the coefficients may be much higher. The user is not limited to these formulae or estimates. It should be noted that longitudinal dispersion is of primary importance for short duration releases. Of course, there are some instances where small dispersion coefficient should not necessarily be conser-vative, such as a release on shore with an intake far off shore.

2.48

2.3.3.2 Description of Computational Methods

Equations (2.35) and {2.39) are evaluated by a standard Simpson's rule numerical integration. Several special procedures are initiated, however, to preserve accuracy and efficiency in the computations!

(1) Limits of Integration

The term within the integral sign of Equations (2.35) and (2.39) can be very nearly zero over part of the range 0 to t, therefore contributing little to the integral. Hence, the range for which the term within the integral sign is nonzero (or greater than or specified small number) is computed so that needless integration is eliminated. The new range t 1 to t 2 is

(2.42)

where y is an arbitrarily large number, normally 50. This range guarantees that the term

· . ( [x - u(t - T)] 2 (t )) ex~ - 4E { t - T) - l - T

. x

will be at least exp (-y}. Additional restrictions placed on t 1 and t 2 are

. t > 0 1 -t2 ~ t. t ~ length of discharge funGtion

(2) Calculation of Infinite Series

(2 .43)

(2 .44)

The inifinite series in Equation (2.35) is truncated to M a· (M < 20), m x max -by the computation

2.49

(2 .45)

where y is an arbitrarily large number, typically 50. This number ensures that the term

(2 .46)

will be at least exp (-y).

The values of the term [cos (n~y/B) cos (n~ys/B)] for n = 1 to 20 are stored before the Simpson's rule integration commences because they are not functions of time. If the computation in Equation (2.45) indicates that more than 20 terms in the series are needed, a single image source solution is used. Poor convergence of the series occurs only close to the source. In the region where the far shore has little effect, the lake solution is appropriate.

The concentration in curies/ft3 at some point downstream or down-gradient is computed directly from a known source release measured in curies/ unit time. This computation takes into account dilution as well as radio-active decay. Dilution and travel time may be found from the concentration models with a few additional computation steps, when they are the desired method of reporting.

(1) Travel Time

"Travel time" applies to an average time of transit between the point of release and the point of use in the water body. The greater the dispersion in a water body, the wider the spread of travel times. This ambiguity is usually not an important consideration in bodies such as unidirectional rivers, but it can become significant in better mixed water bodies such as estuaries. Never-theless, an average travel time can usually be defined; e.g., in the case of an estuary, the travel time would be the distance traveled, x, divided by the net downstream (freshwater) velocity, uf,

2.50

(2 .47)

(2) Dilution

The concept of 11 dilution 11 only has meaning for instantaneous releases or steady releases of a constant source strength.

• Instantaneous Release.

' The dilution, DL, for an instantaneous release to surface or ground water can be computed as

where C is the output from the program, ci/ft3, for a substance with an infinite half-life, M is the total curies under the release function curve (printed out as "AREA UNDER PULSE" in program), and V is the volume of the release, ftj.

The half-life should be set to a very large number, e.g. 1010

years.

Any source strength table may be used for dilution calculations pro-viding it is short enough to be considered instantaneous (e.g. 50 seconds). The total area under the release function is arbi-trary. For simplicity, a source strength of 0.02 curies/sec for 50 seconds would give an area under the curve, M, of 1.0 curie.

• Continuous Source

The dilution for a continuous, constant release, after steady state has been attained, can be computed as

2.51

(2 .48)

(2 .49)

where C = the output from the program for a substance with an infinite

half-life, curies/ft3; q = the release flow rate from the plant, ft3/sec; and

Ss = the arbitrary, source strength, curies/second.

Again, the half-life should be set to a very large number, e.g. 1010 yr.

For dilution computations, it is not necessary to know the true value of the source release; e.g., for simplicity, S maybe set equal ' . s to 1.0 curies/sec in the source strength table, lasting 1010 sec.

2.3.4 Program RIVLAK - User Documentation

2.3.4.1 Program Description

Program RIVLAK is written both in ~SIC and FORTRAN language versions and is interactive. Two subroutines, LINT and SOLUT, are used. LINT linearly interpolates a source-strength table. SOLUT calculates concentration at a specified point in time according to whether the river or the lake solution is applied.

(1) Restrictions

• The maximum number of points defining the source strength table is 39.

• River/lake has constant depth.

• Flow is steady.

• The geometry is straight shoreline or straight rectangular river channel.

• Release of contaminant is from a vertical line source.

• River has constant width.

(2) Computation Procedure

Both river and lake solutions are incorporated in the computation. Option

is provided for selecting the river or the lake solution. However, in the

2.52

river solution mode, when poor convergence of the series in Equation (2.33) occurs or when the effects of the far shore are unimportant, the lake solution is applied. The major steps are:

(a) Read input data. (b) Calculate and store cosine terms in series. (c) Calculate limits of integration. (d) Integrate by Simpson's rule.

At this point, criteria are tested for using river or lake solution. (e) Output results. (f) Repeat (a) through (d) until all desired calculations are

completed. '

The computer program listings and flow chart are included in Appendix C.

2.3.4.2 Input Requirements

Input data required include the source-strength table. It is input in a sequential fashion on the terminal. Other data are as follows:

Variable Description D B

u EX EV Nl vs

X2, T2 TS

V2

Depth Width of river Flow velocity Longitudinal dispersion coefficient Lpteral dispersion coefficient Number of divisions of the release pulse Distance of the point source from shore x, y-position downgradient from the source Time after release commences Half-life of radionuclide

2.3.4.3 Program Operation and Output Decription

Unit

ft ft

ft/sec ft2/sec ft2 /sec

ft ft sec sec

After properly "logging in" to the computer and loading the program, the program will begin by prompting the user to input the source-strength table. For example, suppose the radionuclide discharge is as depicted in Figure 2.16. This pulse is reduced to tabular form by linear approximation as shown in

2.53

(..)

I 4 E - 6 1--::c (.!)

z ~ :;:; 4 LI.I

~ => 0 Vl 2

LINEAR APPROXIMATION

6

O~l~~-'-~~-'-~~_..~~__,,~~--''---'-~.__~~....._~___. o 2 4 6 s 10 I 12 14 16

t - CUTOFF TIME, sec

FIGURE 2.16. Arbitrary Source Release

Table 2.3. This table contains 6 entries (up to 39 may be used). The arbi-trary cutoff time is 11 sec. This cutoff time is useful for truncating the curve without having to modify the data entries. If FORTRAN version is used, all numerical data should be entered with an (8F10.0) format, with decimal points. The table and data input are as follows:

(1) INPUT SOURCE STRENGTH TABLE The source strength illustrated in Figure 2.16 and tabulated in Table 2.3 is input in the following manner: (a) ENTER NUMBER OF POINTS AND CUTOFF TIME (SEC)

> 6' 11 (b) ENTER TIME (SEC) AND VALUE (Cl/SEC)

> 0,0.8, 1.1, 1.1, 1.9, 5.5, 3.0, 6.4 ENTER TIME (SEC) AND VALUE (CI/SEC) > 7.1, 3.1, 13, 1.1

2.54

TABLE 2 .3. Linearized Source Release

Point Time. Value Number Sec Curies/Sec

1 0 0.8 2 1.1 1.1

3 1.9 5.5 4 3.0 6.4 4 3.0 6.4 5 7.1 3.1 6 13 1.1

Time and value are entered in pairs separated by commas {BASIC version). Enter 4 pairs of numbers in a line before the next line is used.

(2) The Program at This Point Outputs

AREA UNDER THE PULSE = CURIES CENTROID IN SEC =

These two values are calculated for the area under the radionuclide dis-charge curve depicted in Figure 2.16 and give an approximate check on the source term.

The following two formulae are used:

t AREA {CURIE) =./:. 2 f{T) dT

0

t Time centroid {sec) = J 2 f{T) T dT/AREA

0

(3) ENTER R FOR RIVER SOLUTION, L FOR LAKE SOLUTION > {In FORTRAN version, ENTER 0 FOR RIVER SOLUTION, INTEGER 1 FOR LAKE SOLUTION >.)

2.55

(4) ENTER DEPTH (FT) > Enter depth of the lake, or, ENTER DEPTH (FT), WIDTH (FT) Enter depth and width of the river.

(5) ENTER U(FT/SEC), EX(FT2/SEC), EY(FT2/SEC), Tl/2(SEC)

> Enter stream velocity, u, dispersion coefficients Ex and Ey and half-life of the radionuclide t 112 •

(6) ENTER NO. OF DIVISIONS OF PULSE > Enter integration steps for numerical integration. Typical value is 25.

(7) ENTER Y POSITION OF SOURCE (FT) > Enter distance in ft of source from shore.

(8) ENTER X AND Y (FT) REL. TO (0.0) AND TIME(SEC) > Enter x position downstream (ft), y position from shore (ft), and elapsed time after release commences; i.e., time and place where concentration needs be calculated.

(9) The program will output concentration corresponding to input in 8. CONCENTRATION = CURIES/CU FT ----

(10) The program repeats (8)' and (9) for further computation.

(11) To terminate the run, input an entry with negative x value, e.g., -1, 0, O.

2.3.5 Sample Problem

2.3.5.1 River Model

Calculate the concentrations in a river resulting from the source shown in Figure 2.16. The river has the following properties:

depth = 25 ft width = 500 ft

u = 1 ft/sec t 112 = 5000 seconds

2.56

Conservatively small values of Ex and EY are estimated from Equations (2.40) and (2.41) using n = 0.035; Ex= 11.5 ft2/sec; and EY = 0.45 ft2/sec.

The source is located on the near shore (ys = 0). Concentrations are calculated on the near shore 7000 ft downstream (x = 7000, y = O} for t = 5000, 7000, 10000, and 12000 seconds. The generation of the source table and the output of the program is illustrated in Figure 2.17.

2.3.5.2 Lake Model

Calculate the concentration onshore and downcurrent in a lake resulting from the source shown in Figure 2.16. The lo~e has the following properties:

Depth = 30 ft u = 0.5 ft/sec

t 112 = 10 seconds.

Conservatively small values of Ex and Ey were calculated using Equations (2.40) and (2.41) with n = 0.02: Ex= 3.84 ft2/sec, Ey = 0.149 ft2/sec.

The source is located on shore (ys = O}. Concentrations are calculated 1000 ft downcurrent and onshore (x = 1000, y = 0) for t = 100, 1000, 2000, and 3000 sec. The generation of the source table and program output is illus-trated in Figure 2.18.

2.3.6 Model Confirmation

The transverse mixing characteristics of the Missouri River in the vicin-ity of the Cooper Nuclear Station near Brownville, Nebraska, were investigated using the fluorescent-dye tracer technique. (l4) Rhodamine WT dye, introduced continuously into the plant's once-through circulating water system, was used to simulate the waste heat. Transverse profiles of dye concentration, depth, and velocity were obtained at several cross sections in the 6-mi reach immedi-ately downstream from the plant. The results of the concentration measurements are used in the model confirmation.

2.3.6.l Model Parameters

The application of the model requires river characteristics and dye release rate. Only the results of those portions of the river where measure-ments were available are compared. These are transects 3, 5, 7, 9, and 10 at

2.57

READY RUN RIVLAK

2-D RIVER/NEAR-SHORE LAKE MODELS

INPUT SOURCE STRENGTH TABLE ENTER NUMBER OF POINTS AND CUTOFF TIME

1 6,11 ENTER TIME AND VALUE

? o,o.a,1.1,1.1,1.9,5.5,3.0,6.4 ENTER TIME

RUN RIVL.AK

2-D RIVER/NEAR-SHORE LAKE MODELS

INPUT SOURCE STRENGTH TABLE ENTER NUMBER OF POINTS AND CUTOFF TIME

'i• 6d1 ENTER TIME AND VALUECCI/SEC>

1 o,o.e,1.1,1.1,1.9,5.s,3.o,6.4 ENTER TI ME< SEC> AND VAUJE (CI /SEC>

'P 7.:L,:~.id:h:l.+1 ENTER R FOR RIVER SOLlJTIONr L FOR LAKE SOLUTION 'P L ENTER DEPTH rf 30 ENTER Ur EX, EYCFT-2/SEC>~ T1/2CSEC> 1 o.5,3.e4,o.149,1.0E7 tNTER NO. OF DIVISIONS OF PULSE 1 25 ENTER Y-POSITION OF SOURCE ·r o THANI\ YOU

AREA UNDER PULSE ~ 39+217 CURIES CENTROID IN SECOND - 5+22739

ENTER X ANDY REL. TO (0,0), AND TIME CSEC> 'i' 1.()00, 0 vi()() CONCENTRATION= O CURIES/CU FT

ENTER x AND y REL. TO co.o>, AND TIME

mil~s 531.5, 530.5, 529.0, 527.0, and 526.11. Manning's n is calculated as 0.018 using average velocity u = 6.5 ft/sec, average depths = 13 ft, and slope= 0.00019. The radius of curvatuve at the lower Brownville Bend is estimated at Re= 8882 ft. The radii of curvature at the Langdon and Aspinwall Bends are 6400 ft and 3400 ft respectively as cited.

The lateral dispersion coefficient EY = 12.0 ft2/sec is used.(l4 ) The longitudinal dispersion coefficient Ex is estimated fro~ Equation (2.40) as 22.5 ft2/sec. The river flow rate on the test day was 56,100 cfs. The plant discharge was 1455 cfs. The discharge canal was located at the right bank of the Missouri River. The dye concentration was nearly uniform in the canal, and the average concentration of 35.6 ppb was close to the theoretical concentra-tion of 36.8 ppb. The canal discharged into the river as a point source and resulted in a fully mixed concentration of 0.95 ppb in the river. The constant rate of discharge of the dye was maintained from 0800 hr until 1333 hr.

2.3.6.2 Model Results and Comparisons

The average river width is taken to be the constant river width required by Program RIVLAK. Using the averaged values to represent a straight channel conforms to the restrictions placed on the model. The results are shown on Figures 2.19a and 2.19b. The agreement with the distributions of relatively near-field transects is poor, because the unweighted relative distance z/W applied as an independent variable in RIVLAK model (as opposed to the flux-weighted relative distance qc/QR applied in the STTUBE model) does not provide an accurate description of the actual mixing process in nonuniform, meandering channels such as the Missouri River. However, further downstream (transects 7, 9, and 10) the agreement is better, because the cumulative effect of Ey tends to average out over the entire length of the test reach.

2.60

6 6

5 TRANSECT 3

4 MILE 531.5 w = 691 ft

3 ·-0--BY MODEL RIVLAK 3 • 2

--MEASURED DATA 2 .0

C>.. C>..

z: l l z: 0 I- 0 0

4 4 TRANSECT 7 -

REFERENCES

1. Yotsukura, N., and E. D. Cobb, "Transverse Diffusion of Solutes in Natural Streams, 11 U.S. Geological Survey, Professional Paper 532-C, 1972.

2. Jackman, A. P., and N. Yotsukura, "Thermal Loading of Natural Streams, 11 U.S. Geological Survey, Open File Report, 75-488, January 1976.

3. Sayre, W.W., and F. M. Chang, 11 A Laboratory Investigation of Open-Channel Dispersion Processes for Dissolved, Suspended, and Floating Dispersants, 11 U.S. Geological Survey, Professional Paper 433-E, 1968.

4. Yotsukura, N., "A Two-Dimensional Temperature Model for a Thermally Loaded River with Steady Discharge," Proceedings, 11th Annual Environmental and Water Resources Engineering Conference, Vanderbilt Univerisity, Nashville, Tennessee, 1972.

5. Bauer D. P., and N. Yotsukura, 11A Two-Dimensional Excess Temperature Model for a Thermally Loaded Stream, 11 U.S. Geological Survey, Gulf Coast Hydroscience Center~ Bay St. Louis, Mississippi, 1974.

6. Henderson, F. M., Open Channel Flow, Macmillan Company, New York, 1959.

7. Fischer, H.B., 11 The Effects of Bends on Dispersion in Streams, 11 Water Resources Research 5{2}, 1969, pp. 496-506.

8. USNRC Reg Guide 1.113, "Estimating Aquatic Dispersion of Effluents from Accidental and Routine Factor Releases for the Purpose of Implementing Appendix I , 11 U.S. Nuclear Regulatory Commission, Revision 1, April 1977.

9. Chow, V. T., Open Channel Hydraulics, McGraw-Hill Book Company, New York, 1959.

10. Sium, O., "Transverse Flow Distribution as Influenced by Cross-Sectional Shape, 11 M. S. Thesis, University of Iowa, Iowa City, Iowa, 1975.

11. Yotsukura, N., W. W. Sayre, "Transverse Mixing in Natural Streams, 11 Water Resources Research, 12(4), August 1976.

12. Yotsukura, N., H. B. Fischer, and W. W. Sayre, "Measurements of Mixing Characteristics of the Missouri River between Sioux City Iowa and Platsmouth Nebraska," ·u.s. Geological Survey, Water Supply Paper 1899-G, 1970.

13. Elder., J. W., 11 The Dispersion of Marked Fluid in Turbulent Shear Flow, 11 Journal of Fluid Mechanics, No. 5, 1959, pp. 544-60.

2 ~63

14. Sayre, W.W., and T. P. Yeh, "Transverse Mixing Characteristics of the Missouri River Downstream from the Cooper Nuclear Station," IIHR Report No. 145, Iowa Institute of Hydraulic Research, April 1973.

15. Csanady, G. T., Dispersal of Effluents in the Great Lakes, Water Research, 1970, pp. 79-114.

16. Chang, Y. C., "Lateral Mixing in Meandering Channels," Ph.D. Thesis presented to the University of Iowa, Iowa City, Iowa, 1971, 195 pp.

2.64

3.0 GROUNDWATER MODELS

3.1 GENERAL DEVELOPMENT

3.1.1 Introduction

The Nuclear Regulatory Commission (NRC) staff has developed straightfor-ward groundwater models for evaluating the transport of radionuclides in groundwater. These models have proven to be useful for estimating the migra-tion of radioactivity from low-level waste facilities, as well as for evaluting the potential contamination of groundwater and surface water from normal and accidental releases from nuclear facilities in general.

The models were developed for uniform, unidirectional flow undisturbed by sources or sinks. This assumption could be violated in the vicinity of highly developed well fields (withdrawals) or injection system (recharge).

The primary criteria in developing the models have been that they (1) Should be relatively easy.to use; (2) Should not require excessive data input and setup; (3) Should not require inordinate amounts of computer time for solution;

and (4) Should give reasonable or conservative estimates.

Such objectives are desirable because (1) Available data often do not warrant the use of complicated models; (2) Sor every long-term (hundred to thousand of years) analysis, such as

encountered for radioactive waste facilities, complicated finite difference or finite element computer models would be too costly to run; and

(3) Simpler models frequently give results adequate for regulatory purposes.

Two programs are developed in Chapter 3.0. The first program, GROUND, presented in section 3.2, calculates concentration and flux from a variety of

sources for a single radioactive component. This program is interactive. The second program, GRDFLX, developed in section 3.3, calculates concentration and flux from sources typical of low-level waste burial grounds with large numbers

of radionuclide